The ancient Greeks transformed trigonometry into an ordered
science. Astronomy was the driving force behind advancements in
trigonometry. Most of the early advancements in trigonometry were in
spherical trigonometry mostly because of its application to
astronomy. The three main figures that we know of in the development
of Greek trigonometry are Hipparchus, Menelaus, and Ptolomy. There
were likely other contributors but over time their works have been
loss and their names have been forgotten.

"Even if he did not invent it, Hipparchus is the first person
of whose systematic use of trigonometry we have documentary evidence."
(Heath 257) Some historians go as far as to say that he invented
trigonometry. Not much is known about the life of Hipp archus. It is
believed that he was born at Nicaea in Bithynia. (Sarton 285) The
town of Nicaea is now called Iznik and is situated in northwestern
Turkey. Founded in the 4th century BC, Nicaea lies on the eastern
shore of Lake Iznik. He is one of the g reatest astronomers of all
time. We know from Ptolemy's references that he made astronomical
observations from 161 to 127 BC. (Sarton 285) Unfortunately, nearly
all of his works are lost, and all that remains is his commentary on the
Phainomena of Eudoxos of Cnidos, and a commentary on an
astronomical poem by Aratos of Soloi. (Sarton 285)
Most of
what we know about Hipparchus comes from Ptolemy's Almagest and a few
remarks by other writers. The sole trigonometric function used by the ancient
Greeks is the chord, which is closely related to the sine function
(Toomer 7).
What is known from Ptolemy is that Hipparchus produced a
table of chords, which were an essential tool in the early
development of trigonometry. According to Theon of Alexandria, who worked in
Alexandria as a teacher of mathematics and astronomy, Hipparchus wrote
a treatise in twelve books on chords in a circle,
which has been lost (Sarton 286) . It is believed that this treatise
contained some general trigonometric theory together with some tables.

Hipparchus is believed to be the first person to determine exactly the
times of the rising and setting of the zodiacal signs. Pappus of
Alexandria, who was a teacher of mathema tics in the fourth century,
observed that "Hipparchus in his book on the rising of the twelve
signs of the zodiac shows by means of numerical calculations that
equal arcs of the semicircle beginning with Cancer which set in times
having a certain relation
to one another do not everywhere show the same relation between the
times in which they rise."(Heath 257) Other mathematicians and
astronomers of the time including Euclid, Autolycus, and Theodosius
could only prove that the times are greater or less in
relation to one another; they could not calculate the actual times.
(Heath 257-258). "As Hipparchus proved corresponding propositions by
means of numbers, we can only conclude that he used propositions in
spherical trigonometry, calculating arcs from ot hers which are given,
by means of tables."(Heath 258).

For his astronomical work Hipparchus
needed a table of trigonometric ratios. It is believed that he
computed the first table of chords for this purpose.
He considered every triangle as being inscribed
in a circle, so that each side became a chord.
While chords were easy to calculate
in some special cases with Euclidean knowledge,
in order to complete
his table Hipparchus would have
needed to know many formulas of plane trigonometry
that he either derived himself or borrowed from elsewhere. Hipparchus
is credited as generalizing Hypsicles' idea of dividing the ecliptic
into 360 degrees, an idea borrowed from the Babylonian astronomers,
by dividing every circle into 360 degrees (Sarton
287).
He divided
the diameter into 120 units and expressed quantities smaller than
degrees as sexagesimal fractions (Sarton 287), in the Babylonian style.

After Hipparchus the next Greek mathematician known to
have made a contribution to trigonometry was Menelaus. We know very little
about the life of Menelaus. Ptolemy mentions that Menelaus
observed in Rome in the year 98 AD (Toomer). Thus it is believed that he
was born around 70 AD (History of Mathematics). Both Pappus and
Proclus call him Menelaus of Alexandria (Heath 260), so we may
assume that he spent some of his time in Rome, and much of his time in
Alexandria. He wrote a six-book treatise on chords, which was
mentioned by Theon of Alexandria, but those books have all been lost.
(Heath 260) His only surviving work is a three-book work called
Sphaerica, whose
third book contains some excellent information about
the development of trigonometry and is the earliest surviving work on
spherical trigonometry. Unfortunately the Greek version of this text
is lost, and
all that remains is an Arabic version translated a thousand years
after the original was written. To make matters worse, various
translators over the
years have had their commentary included in the work, and it becomes
difficult to separate the original from the commentators.
Nevertheless, this work still provides a good source for the development of
Greek trigonometry.

In the first book of the Sphaerica, there is the first known
conception and definition of a spherical triangle (Heath 262).
Menelaus describes a spherical triangle as the area included by arcs
of great circles on the surface of a sphere subject to the restriction
that each of the sides or legs of the triangle is an arc less then a
semicircle. He then goes on to give the main propositions about
spherical triangles corresponding to Euclid's propositions about plane
triangles. (Heath 263). The second book has astronomical interest
only. The third book contains trigonometric ratios. The first
proposition in the third book is Menelaus's theorem with reference to
a spherical triangle and any transversal (great circle) cutting the
sides of a triangle. R ather than using a spherical triangle he
expresses his proposition in terms of two intersecting great circles.
"Between two arcs ADB, AEC of great circles are two other arcs of
great circles DFC and BFE which intersect them and also intersect each
other in F. All the arcs are less than a semicircle."(Heath 266). He
then goes on to prove

which is Menelaus's theorem for spherical trigonometry. In Menelaus'
proof he distinguished three or four separate
cases. Below is a diagram of
Menelaus's theorem for
plane trigonometry:

The rest of the third book consists of trigonometric propositions that
were necessary for astronomical work. The last great contributor to
trigonometry in the Greek period is Ptolomy. Very little is known
about Ptolemy's actual life. He made astronomical observations from
Alexandria in Egypt during the years AD 127-41. The first observation
which we can date ex actly was made by Ptolemy on 26 March 127 while
the last was made on 2 February 141. There is no evidence that
Ptolemy was anywhere other than Alexandria. Heath says "it is evident
that no part of the trigonometry, or of the matter preliminary to it,
in Ptolemy was new. What he did was to abstract from earlier
treatises, and to condense into the smallest possible space, the
minimum of propositions necessary to establish the methods and
formulas used." (276) Other math historians believe that Ptolemy com
pleted the work started by Hipparchus that he worked out some
necessary details and compiled new tables. It is difficult to tell
what additions and modifications Ptolemy made to already existing
works. Toomer calls the Almagest a masterpiece of clarity and method,
superior to any ancient scientific textbook and with few peers from
any period. But it is much more than that. Far from being a mere
compilation of earlier Greek astronomy, as it is sometimes described,
it is in many respects an original work.

Whatever the case, Ptolemy's Almagest is our main source of
information on Hipparchus and on Alexandrian trigonometry. "The
encyclopedic nature of the Almagest, its superior value, and its
formal perfection were probably the main causes of the loss of
Hipparchus' original writings. The early copyists must have felt that
the Almagest rendered previous writings obsolete and superfluous."
(Sarton 286). The use of the Sine, cosine, and tangent functions lay
several hundred years in the future. However, the table of chords can
be used in formulas that are equivalent to present day formulas for
the trigonometric functions. The table of chords in the Alma gest is
likely the same as Hipparchus' table or an expansion of it but we
cannot be sure since we don't have a copy of Hipparchus' table to
compare it with. (Heath 259) Ptolemy's table of chords is completed
for arcs subtending angles increasing from 1/2
degrees to 180 degrees by steps of 1/2 degrees. In order to have
calculated is table of chords Ptolemy must have been aware of the
equivalents of several trigonometric identities and formulas. Ptolemy
was aware of the of the formula, (chord 2x) + (chord (180x - 2x)) =
4r, which is equivalent to sin x + cos x = 1 . Ptolemy also used a
formula that later became known as Ptolemy's theorem. That formula is
chord (a-b) = 1/2 (chord a chord (180-b)) - (chord b chord (180-a))
where a and b are angles. "Pt olemy must have carried out his
calculations to five sexagesimal places to achieve the accuracy he
does in the third place."(Toomer 57-58). Ptolemy's calculations are
accurate enough to be useful today. Here is a partial table of
Ptolemy's chords taken from Toomer:

Ptolemy's Table of Chords

Arcs

Chords

Sixtieths

1/2

0 31 25

1 2 50

1

1 2 50

1 2 50

1 1/2

1 34 15

1 2 50

2

2 5 40

1 2 50

2 1/2

2 37 4

1 2 48

3

3 8 28

1 2 48

3 1/2

3 39 52

1 2 48

4

4 11 16

1 2 47

4 1/2

1 2 47

4 42 40

5

5 14 4

1 2 46

5 1/2

5 45 27

1 2 45

6

6 16 49

1 2 44

6 1/2

6 48 11

1 2 43

7

7 19 33

1 2 42

7 1/2

1 2 41

7 50 54

8

8 22 15

1 2 40

8 1/2

8 53 35

1 2 39

9

9 24 54

1 2 38

9 1/2

9 56 13

1 2 37

10

10 27 32

1 2 35

10 1/2

10 58 49

1 2 33

11

11 30 5

1 2 32

11 1/2

12 1 21

1 2 30

12

12 32 36

1 2 28

12 1/2

13 3 50

1 2 27

13

13 35 4

1 2 25

13 1/2

14 6 16

1 2 23

14

14 37 27

1 2 21

14 1/2

15 8 38

1 2 19

15

15 39 47

1 2 17

...........

.........

.........

............

............

............

Arcs

Chords

Sixtieths

.............

..............

............

............

............

............

165 1/2

119 2 26

0 7 48

166

119 6 20

0 7 31

166 1/2

119 10 6

0 7 15

167

119 13 44

0 6 59

167 1/2

119 17 13

0 6 42

168

119 20 34

0 6 26

168 1/2

119 23 47

0 6 10

169

119 26 52

0 5 53

169 1/2

119 29 49

0 5 37

170

119 32 37

0 5 20

170 1/2

119 35 17

0 5 4

171

119 37 49

0 4 48

171 1/2

119 40 13

0 4 31

172

119 42 28

0 4 14

172 1/2

119 44 35

0 3 58

173

119 46 35

0 3 42

173 1/2

119 48 26

0 3 26

174

119 50 8

0 3 9

174 1/2

119 51 43

0 2 53

175

119 53 10

0 2 36

175 1/2

119 54 27

0 2 20

176

119 55 38

0 2 3

176 1/2

119 56 39

0 1 47

177

119 57 32

0 1 30

177 1/2

119 58 18

0 1 14

178

119 58 55

0 0 57

178 1/2

119 59 24

0 0 41

179

119 59 44

0 0 25

179 1/2

119 59 56

0 0 9

180

120 0 0

0 0 0

The table of chords is equivalent to a table of sines for all central
angles 0 degrees to 90 degrees at 15' intervals and thus can be used
to solve any planar triangle, provided that at least one side is known.
The function sin x is equivalent to 1/2 (chord 2x), and cos x is equivalent to
1/2 chord(180-2x). The Almagest
also contains trigonometric theorems
equivalent to the present day law of sines and the compound-angle and
half-angle identities. The assumption is that Hipparchus also must
have known of these and possibly invented them.

Both Heath and Neugebauer have suggested that the beginnings
of trigonometry as an ordered science go back a few years before
Hipparchus. "The earliest preserved evidence for the approach to
specifically trigonometric problems is found in the treatise, On the
Sizes and Distances of the Sun and the Moon by Aristarchus, written
about 250 BC"(Neugebauer 773). Aristarchus made use of one important
inequality, which is the equivalent of the inequalities
Sin x < x < tan x

With the help of such inequalities Aristarchus estimated the numerical values
of trigonometric functions in some specific cases of small angles. A
few decades later, Archimedes made use of the same formula. al-Biruni
has preserved a Lemma of Archimedes, which shows
that he had an equivalent version of Ptolemy's Theorem at his
disposal (Neugebauer 773). In Menelaus' work there is a remark that
suggests that one of the trigonometric propositions can be attributed
to Apollonius, who lived a few years before Hipparchus
(Heath 253). "Tannery (from his Recherches sur l'hist. De
l'astronomic ancienne, p. 64) ... suggested that not only Apollonius
but Archimedes before him may have compiled a table of chords or at
the least shown the way to such a compilation." (Heath 253)

As we have seen, the beginnings of trigonometry can be traced far
back into history. Hipparchus, in the middle of the first century BC,
was the first person known to have treated trigonometry as an applied
science, and the first person to compile a table of chords. Menelaus
greatly advanced the field of spherical trigonometry.
"By the second century A.D. trigonometry had reached its final form
before Islamic developments in Book I of the Almagest"
(Neugebauer, p. 772).